# Can AIC/BIC be used to compare models that differ by transformation of response variable? [duplicate]

I fit two models: one with a log-transformed response variable, and one without. All else is the same between them. Can I use AIC (or BIC or other, similar criteria) to choose which approach best fits the data?

No it should not be used.

The Akaike Information Criterion (AIC) and the Bayes Information Criterion (BIC) are derived from the log-likelihood $$\cal{L}$$ of the model:

$$\text{AIC}= -2\cal{L}+ 2p$$ $$\text{BIC}= -2\cal{L} + p\log(n)$$

where p is number of predictors in model (degrees of freedom), n the number of observations. The -2 loglikelihood (also known as the deviance) term above is normally dependent on the dependent variable

For example in linear regression models, it is $$n\log(\text{RSS}/n)$$, where $$\text{RSS}$$ is the sum squared of residuals:

$$\text{RSS}=\sum_{i=1}^n (y_i-\hat{y_i})^2$$

We can look at an example, in R, where below, both logged response and original scale give, i would say, equally good fit, but differs in BIC. If one uses BIC or AIC it will be misleading

fit = lm(carb ~.,data=mtcars)
fit_log = lm(log(carb) ~.,data=mtcars)

BIC(fit)
[1] 95.80717
BIC(fit_log)
[1] 30.53078

par(mfrow=c(1,2))
plot(fitted(fit),mtcars$$carb) mtext(paste("cor=",round(cor(fitted(fit),mtcars$$carb),digits=3)))
plot(fitted(fit_log),log(mtcars$$carb)) mtext(paste("cor=",round(cor(fitted(fit_log),log(mtcars$$carb)),digits=3)))